Quantum field theory 2, lecture 06

Screened potential

In non-relativistic physics, the propagation of photons is so fast that it is effectively instantaneous. The photon becomes then a non-dynamical field. If \(\mathbf{p}\) is a wave-vector measured in units of inverse length, and \(p^0\) is a frequency, the inverse photon propagtor becomes in the limit where the velocity of light is large \[-\frac{1}{c^2} (p^0)^2 + \mathbf{p}^2 + m^2 \to \mathbf{p}^2 + m^2.\] Note that in this counting \(m\) has units of inverse length. Accordingly the potential between two unit changes in position space becomes \[\begin{split} V(\mathbf{x}-\mathbf{y}) = & \int \frac{d^3 p}{(2\pi)^3} e^{i\mathbf{p}(\mathbf{x}-\mathbf{y})} \frac{1}{\mathbf{p}^2+m^2} \\ = & \frac{1}{4\pi^2} \int_0^\infty dp \int_{-1}^1 d(\cos(\vartheta)) \frac{e^{i\cos(\vartheta)p|\mathbf{x}-\mathbf{y}|}p^2}{p^2+m^2} \\ = & \frac{1}{4\pi^2} \int_0^\infty dp \frac{p e^{i p|\mathbf{x}-\mathbf{y}|} - p e^{-i p|\mathbf{x}-\mathbf{y}|}}{(p^2+m^2)i |\mathbf{x}-\mathbf{y}|} \\ = & \frac{1}{4\pi^2} \int_{-\infty}^\infty dp \frac{p e^{i p|\mathbf{x}-\mathbf{y}|}}{(p^2+m^2)i |\mathbf{x}-\mathbf{y}|} \\ = & \frac{e^{-m|\mathbf{x}-\mathbf{y}|}}{4\pi |\mathbf{x}-\mathbf{y}|}. \end{split}\] (The last integral has been done by closing the integral in the upper half of the complex plane and using the residue theorem.) What would be the Coulomb potential for \(m=0\) has now become a Yukawa potential with screening length \(1/m\). The electromagnetic interaction becomes a short range interaction! This has a direct consequence for superconductors, where magnetic fields are quickly decaying inside the superconducting material (Meissner effect). In this sense, the Meissner effect can be seen as a variant of the Higgs mechanism for electromagnetism!

In a similar way to what we have described here, the gauge bosons \(W^{\pm}\) and \(Z\) in the electroweak standard model gain their mass and the weak nuclear forces becomes short-range.

Photon “eats” Goldstone boson

Massless photons have two internal degrees of freedoms, namely two polarizations or helicities. Massive spin-one particles on the other hand have three internal degrees of freedom, namely their three spin states. Where does this come from? Spontaneous symmetry breaking of a global continuous symmetry produces a Goldstone boson, while the breaking of a local continuous symmetry results in a massive photon, but no Goldstone boson.

Sometimes it is convenient to reparameterize the complex scalar field \(\Phi(x)\) in a non-linear way. One such non-linear parameterization is \[\Phi(x)=\sigma(x) e^{i \pi(x)},\] where \(\sigma(x)\) and \(\pi(x)\) are real scalar fields. In the low energy effective action of QCD, \(\sigma(x)\) corresponds to the sigma resonance (which is very broad and difficult to observe) and \(\pi(x)\) is an analog of the pion. In this parameterization, the gauge transform shifts the pion field \(\pi(x)\to\pi(x)+\alpha(x)\), while \(\sigma(x)\) is invariant under electromagnetic gauge transformations.

We will now express the effective action in terms of the new fields. First, the effective potential does not depend on \(\pi(x)\) at all, \(U(\Phi^*\Phi)=U(\sigma^2)\). For a global \(\text{U}(1)\) symmetry, \(\pi(x)\) is the Goldstone boson – it is a massless excitation with no potential term and only the kinetic term in the action. In contrast, \(\sigma(x)\) is the radial mode and has a non-vanishing mass \[m^2= \frac{\partial^2 U(\sigma^2)}{\partial \sigma^2} = 2 \lambda \rho_0.\]

Rewrite action in terms of \(\sigma(x)\) and \(\pi(x)\)

Now we rewrite covariant derivaties of the complex field, \[\begin{split} D_\mu \Phi(x) = & [\partial_\mu - i e A_\mu(x)] \sigma(x) e^{i\pi(x)} \\ = & \left[\partial_\mu\sigma(x) + i \sigma(x)\left[\partial_\mu\pi(x) - eA_\mu(x) \right] \right] e^{i\pi(x)}. \end{split}\] The kinetic term in the action becomes accordingly \[\begin{split} D^\mu \Phi^*(x) D_\mu \Phi(x) = & \partial^\mu \sigma(x) \partial_\mu\sigma(x) \\ & + \sigma(x)^2 \left[ \partial^\mu\pi(x) - e A^\mu(x) \right] \left[ \partial_\mu\pi(x) - e A_\mu(x) \right]. \end{split}\] We can check that gauge invariance still holds. If \(\pi(x)\to\pi(x)+\alpha(x)\) then \(\partial_\mu \pi(x)\to\partial_\mu \pi(x)+\partial_\mu \alpha(x)\), while \(eA_\mu(x)\to eA_\mu(x) +\partial_\mu \alpha(x)\). Indeed the gauge symmetry is conserved since the pion field only appears in the combination \(\partial_\mu \pi(x)-eA_\mu(x)\)! Actually we can use the gauge symmetry to make the pion field constant \(\partial_\mu\pi(x)=0\), and even vanishing, \(\pi(x)=0\). This is called unitary gauge. Then \(\pi(x)\) disappears from the quantum effective action and the resulting field equations. One says that the photon “eats” the Goldstone boson and becomes massive and we are left with the action \[\Gamma[\sigma,A]=\int_x \left\{ \partial^\mu \sigma \partial_\mu \sigma + \frac{1}{4}F^{\mu\nu}F_{\mu\nu}+e^2 \sigma^2 A^\mu A_\mu + U(\sigma^2) \right\} .\] It describes an effectively massive gauge field and a massive scalar.

Gauge redundancy

Local gauge theories are “redundant” descriptions. For every generator of the gauge group, there is one degree of freedom on which nothing depends. It can be eliminated by gauge fixing. Different gauge fixings eliminate different fields. For electromagnetism, one may choose Landau gauge \(\partial_\mu A^\mu = 0\) which eliminates longitudinal photons or, in the presence of a broken \(\text{U}(1)\) symmetry unitary gauge \(\pi(x)=0\), which eliminates the Goldstone boson. Of course one cannot apply both conditions simultaneously. Gauge fixings are physically equivalent, even though the gauge fixed actions might look different.

A good reason to accept a redundant description is the locality of the gauge covariant action, which would get lost if one would attempt to elliminate all gauge degrees of freedom.

Electroweak symmetry breaking

A very similar phenomenon occurs for the spontaneous breaking of the electroweak gauge symmetry group \(\text{SU}(2)\times \text{U}(1)\) to the electromagnetic gauge symmetry group \(\text{U}(1)\). The standard model invovles a complex scalar doublet \[\Phi(x) = \begin{pmatrix} \varphi_1(x)+i\varphi_2(x) \\ \varphi_3(x)+i\varphi_4(x) \end{pmatrix},\] and four gauge bosons: a triplet \(\mathbf{W}_\mu(x)\) for the three generators of \(\text{SU}(2)\) and a singlet \(Y_\mu(x)\) for \(\text{U}(1)\). The electroweak fields \(W^\pm_\mu(x)\) and \(Z_\mu(x)\) as well as the electromagnetism gauge field \(A_\mu(x)\) are linear combinations of \(\mathbf{W}_\mu(x)\) and \(Y_\mu(x)\). The symmetry \(\text{SU}(2)\times \text{U}(1)\) is broken down to \(\text{U}(1)\) by an expectation value, which can be brought to the form \[\langle \Phi(x) \rangle =\begin{pmatrix} \varphi_0 \\ 0 \end{pmatrix}.\] The \(W^\pm\) and \(Z\) bosons acquire mass through the Higgs mechanism but the photon remains massless. The real scalars \(\varphi_2(x)\), \(\varphi_3(x)\) and \(\varphi_4(x)\) disappear from the spectrum, much like \(\pi(x)\) in the Abelian model. In contrast, \(\varphi_1(x)\) plays the role \(\sigma(x)\) played before. It descibes a massive scalar parrticle in the low energy description. This is the Higgs boson which has been found at the Large Hadron Collider at CERN.

Saddle point approximation and perturbation theory

In this chapter we will start a computation of the quantum effective action \(\Gamma_\text{E}[\Phi]\) in the Euclidean domain. We assume that interactions are small, and that some type of perturbation expansion in the small couplings should be possible. We recall that in the absence of interactions the microscopic action \(S_\text{E}[\phi]\) is quadratic in \(\phi\), and we have shown that then \(\Gamma_\text{E}[\Phi]=S_\text{E}[\Phi]+\text{const}\), where the constant part can actually depend on external parameters like temperature or chemical potential, or external fields like the metric \(g_{\mu\nu}(x)\) or an extneral gauge field \(A_\mu(x)\).

Background field identity

Our starting point is the functional integral representation of \(\Gamma[\Phi]\), or background field identity, in the Euclidean domain, using abstract index notation, \[\Gamma_\text{E}[\Phi]=-\ln\int D\phi^\prime \exp\left( -S_\text{E}[\Phi+\phi']+ \frac{\delta \Gamma_\text{E}[\Phi]}{\delta \Phi^\alpha}\phi^{\prime\alpha} \right),\] where \(\Phi^\alpha\) is the background field or expectation value, and \(\phi^{\prime\alpha}\) is the fluctuation field. We separate the classical contribution, \[\Gamma_\text{E}[\Phi]=\underbrace{S_\text{E}[\Phi]}_\text{classical contribution}\underbrace{-\ln\int D\phi^\prime \exp\left( -S_\text{E}[\Phi+\phi']+S_\text{E}[\Phi]+\frac{\delta \Gamma_\text{E}[\Phi]}{\delta \Phi^\alpha}\phi^{\prime\alpha} \right) }_\text{fluctuation contribution}.\] This is similar to the free energy: there are energy and entropy contributions. Every term beyond the classical action is called “fluctuation contribution” or “loop contribution”, \[\Gamma_\text{E}[\Phi]=S_\text{E}[\Phi]+\Gamma_\text{E, loops}[\Phi].\] Here \(\Gamma_\text{E, loops}[\Phi]\) accounts for all loops in perturbation theory. Note that the expression for \(\Gamma_\text{E, loops}[\Phi]\) is implicit because \(\delta \Gamma_\text{E}[\Phi]/\delta \Phi^\alpha\) appears on the right hand side.

Saddle point expansion

We expand \(S_\text{E}[\Phi+\phi^\prime]\) around \(\phi^\prime=0\), \[S_\text{E}[\Phi+\phi^\prime]=S_\text{E}[\Phi]+\frac{\delta S_\text{E}[\Phi]}{\delta \Phi^\alpha}\phi^{\prime\alpha}+\frac{1}{2} \frac{\delta^2 S_\text{E}[\Phi]}{\delta\Phi^\alpha\delta\Phi^\beta}\phi^{\prime\alpha}\phi^{\prime\beta}+\ldots\] The first derivative term \(\delta S_\text{E}[\Phi]/\delta \Phi^\alpha\) cancels against the classical term in \(\delta \Gamma_\text{E}[\Phi]/\delta \Phi^\alpha\), \[\Gamma_\text{E, loops}[\Phi]=-\ln \int D \phi^\prime \exp\left( -\frac{1}{2} \frac{\delta^2 S_\text{E}[\Phi]}{\delta\Phi^\alpha\delta\Phi^\beta}\phi^{\prime\alpha}\phi^{\prime\beta} +\ldots + \frac{\delta \Gamma_\text{E, loops}[\Phi]}{\delta \Phi^\alpha}\phi^{\prime\alpha} \right) .\] We can now proceed to an iterative solution. The lowest order is the one-loop approximation. Here one neglects \(\delta \Gamma_\text{E, loops}[\Phi]/\delta \Phi^\alpha\) and higher order terms in the expansion, like \(S_\text{E}^{(3)}[\Phi]\) etc. What remains is a Gaussian integral, \[\begin{split} \Gamma_\text{E, 1-loop}[\Phi] = & -\ln\int D\chi^\prime\exp\left( -\frac{1}{2}\frac{\delta^2 S_\text{E}[\Phi]}{\delta\Phi^\alpha\delta\Phi^\beta}\phi^{\prime\alpha}\phi^{\prime\beta} \right)\\ = & -\ln\left(\text{Det}\left(S_\text{E}^{(2)}[\Phi]\right)^{-1/2}\right) + \text{const} \\ = & \frac{1}{2}\text{Tr}\left\{\ln (S_\text{E}^{(2)}[\Phi])\right\}+\text{const}. \end{split}\] Here we used the identity \[\ln(\text{Det}(M)) = \text{Tr}\{\ln(M)\},\] The logarithm can only be taken in a basis where \(M\) is diagonal. The trace Tr is in the sense of operators and includes here contiuous and discrete indices.

Remarks

• The saddle point approximation requires \(S_\text{E}^{(2)}[\Phi]\) to be positive semidefinite (to have positive or at most vanishing eigenvalues). Then the Gaussian integral for a Euclidean functional integral is well defined.

• The second functional derivative of the classical action \(S_\text{E}^{(2)}[\Phi]\) is the inverse of the classical Euclidean propagator in the presence of a background field \(\Phi_j(x)\).

• The functional trace is sometimes difficult to evaluate, specifically for inhoogeneous background fields. Special mathematical methods such as the heat kernel expansion have been depeloped for this purpose.

Evaluation in momentum space

In momentum space the 1-loop effective action can be evaluated as \[\Gamma_\text{E, 1-loop}[\Phi]=\frac{1}{2}\underbrace{\int\frac{d^{d}p}{(2\pi)^{d}}\,\int\frac{d^{d}q}{(2\pi)^{d}}\, (2\pi)^d \delta^{(d)}(p-q)}_\text{Tr} (\ln \det S_\text{E}^{(2)}[\Phi])(p,q).\] Here \(\det\) stands for the determinant in the space of discrete field components labed by the indices \(j\) and \(k\). For example take \(\Phi_n(x)=\Phi_n\) to be constant in space, then the inverse microscopic propagator becomes diagonal in momentum space, \[(S_\text{E}^{(2)})_{jk}(p,q)[\Phi] = P_{jk}(p,\Phi) (2\pi)^d \delta^{(d)}(p-q).\] We need to set here \(p=q\) and integrate over it. Using now also that \[(2\pi)^{d} \delta^{(d)}(0) = \int_x,\] which is just the volume of space-time, one finds \[\Gamma_\text{E, 1-loop}[\Phi] = \int_x U_\text{1-loop}(\Phi)\] with the one-loop contribution to the effective potential \[U_\text{1-loop}(\Phi)=\frac{1}{2} \int\frac{d^d p}{(2\pi)^d} \ln \det P(p, \Phi).\] We will investigate this expression in more detail below.

One loop effective potential

We employ the generic form of the quantum effective action \(\Gamma[\Phi]\), \[\Gamma_\text{E}[\Phi]=\int_x U(\Phi)+\text{derivative terms}.\] The effective potential \(U(\Phi)\) depends on scalar fields and involves no derivatives. For the computation of \(U(\Phi)\) one evaluates \(\Gamma[\Phi]\) for homogeneous scalar fields. For \(\partial_\mu \Phi=0\) the derivate terms do not contribute.

We investigate first a real scalar field with classical action \[S_\text{E}[\phi]=\int_x \left\{ \frac{1}{2} g^{\mu\nu}\partial_\mu \phi \partial_\nu \phi+V(\phi) \right\}.\] In Euclidean space, and when using cartesian coordinates, the metric is simply \(g_{\mu\nu} = g^{\mu\nu} = \delta_{\mu\nu}\). We take the microscopic potential to be \[V(\phi)= \frac{\bar{m}^2}{2} \phi^2+\frac{\bar{\lambda}}{8}\phi^4.\] With the derivatives \[%\begin{split} \frac{\partial V}{\partial \phi} = \bar{m}^2\phi+\frac{\bar{\lambda}}{2}\phi^3, \quad\quad\quad \frac{\partial^2V}{\partial\phi^2} = \bar{m}^2+\frac{3}{2}\bar{\lambda}\phi^2, %\end{split}\] one finds in momentum space for \(S_\text{E}^{(2)}[\Phi]\), evaluated at \(\phi=\Phi=\text{const}\), \[S_\text{E}^{(2)}(p,q)[\Phi]=(p^2+\bar{m}^2+\frac{3}{2}\bar{\lambda}\Phi^2) (2\pi)^d\delta^{(d)}(p-q),\] and therefore the inverse propagator function is \[P(p,\Phi)=p^2+\bar{m}^2+\frac{3}{2}\bar{\lambda}\Phi^2.\] The propagator depends now on the field expectation value \(\Phi\)! This yields the one loop contribution to the effective potential \[U_\text{1-loop}(\rho)=\frac{1}{2}\int\frac{d^{d}p}{(2\pi)^{d}}\, \ln(p^2+\bar{m}^2+3 \bar{\lambda}\rho)\] with \(\rho=\frac{1}{2}\Phi^2\). Introducing an explicit ultraviolet regulator scale \(\Lambda\), one finds for the effective potential at this order \[U(\rho)= \bar U_\Lambda + \bar m^2_\Lambda \rho +\frac{1}{2}\bar{\lambda}_\Lambda \rho^2+\frac{1}{2}\int_{p^2<\Lambda^2}\ln(p^2+\bar{m}^2_\Lambda+3 \bar{\lambda}_\Lambda \rho).\] We have introduced also a constant part \(\bar U_\Lambda\) which could be chosen conveniently.